Is the empty set a subset of itself?

There is only one empty set. It is a subset of every set, including itself. Each set only includes it once as a subset, not an infinite number of times.

Let $A$ and $B$ be sets. If every element $a\in A$ is also an element of $B$, then $A\subseteq B$.

Flip that around and you get

If $A\not\subseteq B$, then there exists some element $x\in A$ such that $x\notin B$.

If $A$ is the empty set, there are no $x$s in $A$, so in particular there are no $x$s in $A$ that are not in $B$. Thus $A\not\subseteq B$ can't be true. Furthermore, note that we haven't used any property of $B$ in the previous line, so this applies to every set $B$, including $B=\emptyset$.

(From a wider standpoint, you can think of the empty set as the set for which $x\in \emptyset\implies P$ is true for every statement $P$. For example, every $x$ in the empty set is orange; also, every $x$ in the emptyset is not orange. There is no contradiction in either of these statements because there are no $x$'s which could provide counterexamples.)

The empty set is subset of the empty set, as every element of the empty set is an element of the empty set. But $0$ is not in the empty set.

$A \subseteq B$ when $x\in A \implies x\in B$. As $x\in A \iff x\in A$ we see that $A \subseteq A$ is always true, when $A$ is a set.

A value is a value not a set, sometimes $0$ is defined as the empty set but then $0$ is the empty set and not the number.

This happens for example in category theory, as you are only interested in abstract sets, and all sets of the same cardinality are in a sense the same, you just title finite sets by their cardinality.